Hi BG,
Its not really a change of name Elizabeth has so many variations, Eliza, Betty, Betsy, Beth , Liz, Lizzie the list goes on.

wonder if her illegitimate child could be the son of the Joseph that was also working there
I think thats a distinct possibiity they usually named them after the father when they were illegitimate, I would just keep an open mind until you are able to... if at all to find anything to substanciate this.
t might explain why she named the child Joseph Markham (slight change from Marnham to make it less obvious) - any thoughts on that one???
My answer to that one is the same as before, anything is possible, see if thats any help.

I will try and find out where Balderton is in relation to Caunton.